I'm a 17 years old and I have no clue about a concept known as limit cycles. I looked it up and I understand it represents the orbit of functions approaching other A person told me that limit cycles had something to do with non-linear asymptotes(curved, function-like asymptotes). If you look it up in wikipedia the actual definition is curvilinear asymptotes.
For example if we take a function such as $$\lim_{x\to\infty}\frac{\sqrt{x^3+5x+5}}{x+5}$$. We can use manipulation by limits to get...
$$\lim_{x\to\infty}\frac{\sqrt{x^3+5x+5}}{x+5}$$ $$\sqrt{\lim_{x\to\infty}\frac{x^3+5x+5}{x^2+10x+25}}$$
Then using polynomial division.
$$\sqrt{x-10+\frac{80x+255}{x^2+10x+25}}$$ $$\sqrt{x-10}$$ So... $$\lim_{x\to\infty}\frac{\sqrt{x^3+5x+5}}{x+5}=\sqrt{x-10}$$
Now how do limit cycles explain this? I appreciate any help from people who is mathematically wise to explain this.
You are computing a basic limit, which does not really have anything to do with a "limit cycle."
People usually talk about limit cycles in the context of a dynamic system. An example is the following: Define a function $f(x)$ by:
$$ f(x) = -\arctan(3x) $$
Consider the following dynamical system: Take any nonzero value $x_0$ and define:
\begin{align} &x_1 = f(x_0)\\ &x_2 = f(x_1)\\ &x_3 = f(x_2)\\ &x_4 = f(x_3)\\ \end{align}
and so on, so that $x_{n+1} = f(x_n)$. Try it. Here is what I get with $x_0=1$:
\begin{align} &x_0=1\\ &x_1=-1.249045772398254\\ &x_2=1.310003745871587\\ &x_3=-1.321631768605089\\ &x_4= 1.323735486003704\\ &x_5=-1.324112374293736\\ &x_6=1.324179776191448\\ &x_7=-1.324191826399697\\ &x_8=1.324193980631707 \\ &x_9=-1.324194365742789\\ &x_{10}=1.324194434588798 \end{align}
Here is what I get with $x_0=100$: \begin{align} &x_0=100\\ &x_1= -1.567463005807160 \\ &x_2= 1.361259896088634 \\ &x_3= -1.330650571776570 \\ &x_4= 1.325343336954363 \\ &x_5= -1.324399667758768 \\ &x_6= 1.324231130910750 \\ &x_7= -1.324201006895994 \\ &x_8= 1.324195621816688 \\ &x_9=-1.324194659135894 \\ &x_{10} = 1.324194487038431 \end{align}
It can be shown that there is a special point $x^*>0$ such that $f(f(x^*)) = x^*$, and whenever $x_0\neq 0$ the sequence $\{x_n\}_{n=1}^{\infty}$ approaches the "limit cycle" of alternating between $x^*$ and $-x^*$. In particular, if $x_0>0$ then $\lim_{n\rightarrow\infty} x_{2n} = x^*$ and $\lim_{n\rightarrow\infty} x_{2n+1} = -x^*$. You can say that the dynamics asymptotically approach a "period 2 orbit," even though the sequence never actually repeats itself (unless you use either $x_0=x^*$, $x_0=-x^*$, or $x_0=0$). The case $x_0=0$ yields the "unstable stationary behavior" $x_n=0$ for all $n$ (unstable because if there is any slight deviation in the initial condition, the trajectory will converge to the limit cycle instead of the all-zero behavior).
You also get limit cycles in astronomy: Imagine a comet that shoots near the sun, gets trapped by the sun's gravity, and ends up orbiting the sun in a trajectory that approaches an ellipse.